Vnicity Principles in the Potential Theory*^

Transcription

1 Kishi, Masanori Osaka Math. J. 13 (1961), Vnicity Principles in the Potential Theory*^ By Masanori KISHI Introduction The theory of potentials in a locally compact space has been mainly concerned with potential theoretical principles and the characterization of the class of kernels which satisfy those principles. The principles are numerous and some of them are complicated, but typical ones are the balayage, equilibrium, energy principles and a few others. Roughly speaking, a kernel is said to satisfy the balayage (resp. equilibrium) principles when balayaged (resp. equilibrium) measures exist. The principle states only the existence of balayaged (resp. equilibrium) measures. Naturally the uniqueness of balayaged (resp. equilibrium) measures comes into question and the principle should be split into two principles, the one on the existence and the other on the uniqueness. The present paper is devoted to those unicity principles, which are formulated as follows: (i) if potentials of positive measure coincide with each other in the whole space except for at most a potential theoretical null set, then the measures are identical (^-principle), (ii) the balayaged measure is uniquely determined by a given positive measure and a compact set (B\J-principle) y (iii) the equilibrium measure is uniquely determined by a given compact set (EU-principle). Needless to say, the BU- (resp. EU-) principle is considered as to kernels which satisfy the balayage (resp. equilibrium) principle. By the classical theory it is known that the kernels of order oc (0<^a<^ή) in the w-dimensional Euclidean space R n satisfy the U- principle and if n 2<βί<^n y they satisfy both the BU- and EU-principles, and that the Green kernels on open Riemann surfaces satisfy the three principles. Recently, concerning symmetric kernels which satisfy the balayage principle and a regularity condition, Ninomiya [11] has given a necessary and sufficient condition for them to satisfy the U-principle. From his argument it follows that his condition is sufficient also for *) This has been done under the scholarship of the Yukawa Foundation.

2 42 M. KISHI the BU-principle. He proposed a question which may be expressed as follows : Is it true that his condition is sufficient for the EU-principle? It will be shown that the answer is negative. A characterization of the U-principle was obtained by Anger [1], too. His result states that a kernel satisfies the U-principle if and only if every continuous function with compact support is approximated uniformly by a sequence of potentials of signed measure. In the present paper we consider symmetric kernels and we aim to obtain simple necessary and/or sufficient conditions for the three unicity principles in terms of a kernel itself. In Chapter I, we give necessary conditions for the principles, and we get the relations among the conditions. Each of the following conditions is necessary in order that a symmetric kernel K satisfies one of the unicity principles : (1) K is non-degenerate ( 1), (2) ϋf-potentials separate points ( 1), (3) K satisfies Ninomiya's condition mentioned above ( 2). We remark that the necessity of (3) was proved by Ninomiya only for the U-principle. Condition (3) is stronger than (1), and (1) is stronger than (2). In 3, we show that for symmetric composition kernels (1) and (2) are equivalent, and if they satisfy the balayage or equilibrium principle, then the three conditions are equivalent. We add to them two other necessary conditions for the unicity principles, one of which was obtained by Ninomiya. To obtain sufficient conditions we prepare two approximation theorems in Chapter II. The first is due to Bourbaki [2] and the second is new so far as the author knows. In Chapter III we give sufficient conditions. First we introduce three lower envelope principles the strong, compact and ordinary lower envelope principles, and we obtain the relations among these principles, the domination (or balayage) and equilibrium principles. Deny [6] proved the equivalence between the strong lower envelope principle and the domination principle for his distribution-kernels of positive type, but for our kernels the equivalence fails to hold. If a kernel satisfies the domination principle, then it satisfies the compact lower envelope principle, but not necessarily the strong lower envelope principle, and if a kernel of positive type satisfies the compact lower envelope principle, then it satisfies the domination principle. A sufficient but not necessary condition will be obtained for balayable kernels to satisfy the strong lower envelope principle. Other relations between the lower envelope principle and the equilibrium principle will be added. Next we define "regularity" of kernels. This is an auxiliary notion

3 Unicity Principles 43 by which the approximation theorems mentioned above are available. In the following section we prove that in order that a regular balayable kernel satisfies the U- and BU-principles, "non-degeneracy" is sufficient. From this follows immediately the sufficiency of Ninomiya's condition. The condition "potentials separate points" is not sufficient in general, but for kernels satisfying both the balayage and equilibrium principles it is sufficient. As to regular composition kernels satisfying the balayage principle it is shown that the conditions considered in Chapter I are sufficient. In the last section we show that the condition "potentials separate points" is sufficient for the EU-principle as to kernels satisfying the lower envelope principle. Definitions and known results We shall be concerned with a locally compact space X and positive measures μ on X with compact support S μ. A real-valued continuous function K(x, y) defined on the product space XxX is called a kernel when it satisfies the following conditions: (i) 0<^K(x, jy)<c + and (ii) K(x> y) is finite except for at most the diagonal set of XxX. The kernel K defined by K(x> y) = K(y, x) is called the adjoint kernel. When K^ΞΞΞK, K is called the symmetric kernel. Given a positive measure μ, we define the (K-)potential Kμ{x) by the integral and the adjoint potential Kμ(x) by K μ (x) = fox, y)d μ {y) = fay, x)dμ{y). By Fubini's theorem it holds that \Kμ{x)dv{x) = \Kv(x)dμ(x) for any positive measures μ and v. The integral \Kμ{x)dμ{x) is called the energy of μ. A Borel set BcX is, by definition, of positive capacity, if it contains a compact set F which supports at least a positive measure μφo with finite energy otherwise B is of capacity zero. We say that a property holds nearly everywhere on a Borel set B when the set of points of B at which the property fails to hold is a Borel set of capacity zero. It is easily verified that if a property holds nearly everywhere on Borel oo sets B n (n = l y 2, 3, ), then it holds nearly everywhere on \JB n.

4 44 M. KISHI Now we define potential theoretical principles which will be considered in this paper. THE CONTINUITY PRINCIPLE. If Kμ{x) is finite and continuous on the support S μ, then it is continuous in the whole space. THE EQUILIBRIUM PRINCIPLE. For any compact set F, there exists a positive measure μ supported by F such that Kμ(x) = l nearly everywhere on F, and Kμ{x)<Zl everywhere in X. The measure μ is called the equilibrium measure of F and Kμ{x) the equilibrium potential. THE BALAYAGE (SWEEPING-OUT) PRINCIPLE. For any compact set F and for any positive measure μ, there exists a positive measure v supported by F such that Kv(x) = Kμ{x) nearly everywhere on F and Kv(x)<Kμ(x) everywhere in X. The measure v is called the balayaged measure of μ to F and Kv(x) the balayaged potential. A kernel satisfying the balayage principle is called balayable. FROSTMAN'S MAXIMUM PRINCIPLE. If Kμ{x)<il on the support S μ, then the same inequality holds everywhere in X. THE DOMINATION PRINCIPLE (CARTAN'S MAXIMUM PRINCIPLE). If a potential Kμ(x) of a positive measure with finite energy is dominated by a potential Kv{x) on the support S μ, then Kμ{x) is also dominated by a potential Kv(χ) in the whole space. THE ENERGY PRINCIPLE. This principle is considered only for symmetric kernels K. For any distinct positive measures μ and v with finite energy the expression ) dμ(x) - 2 is finitely determined and ^>0. When the expression (1) is non-negative, the symmetric kernel K is said to be of positive type. THE U-PRINCIPLE. If potentials of positive measure coincide with each other nearly everywhere in the whole space, then the measures are identical. THE EU-PRINCIPLE. The equilibrium measure is uniquely determined by a given compact set. THE BU-PRINCIPLE. The balayaged measure is uniquely determined by a given positive measure and a compact set. Throughout this paper we assume that every open set is of positive capacity, that is, for any open set G, there exists a positive measure μφo such that the support S μ is contained in G and the energy VKμdμ is finite. Concerning symmetric kernels Ninomiya [11] obtained the following theorems which we use throughout this paper.

5 Unicity Principles 45 Theorem A. If K satisfies Frostmaris maximum principle or the domination principle, then it is of positive type. Theorem B. A kernel K satisfies Frostmaris maximum principle if and only if it satisfies the equilibrium principle. Theorem B'. // K satisfies the equilibrium principle, then the equilibrium potentials of a compact set coincide with each other nearly everywhere in X. Theorem C. A kernel K satisfies the domination principle if and only if it satisfies the balayage principle. Theorem C. If K satisfies the balayage principle, then the balayaged potentials of a positive measure to a compact set coincide with each other nearly everywhere in X. Theorem D. In order that K satisfies Frostmaris maximum principle, it is necessary and sufficient that it has the property [PJ. Let x 0 be an arbitrarily fixed point and let λ be a positive measure with compact support S λ which does not contain x 0. If KX(x) <C K(x 0, x) on S λ, then the total measure of λ is <C1. Therem E. In order that K satisfies the domination principle, it is necessary and sufficient that it has the property [P 2 ]. Let x 0 be an arbitrarily fixed point and let λ be a positive measure with compact support S λ which does not contain x 0. If KX(x) <C K(x 0, x) on S λ, then the same inequality holds in X. The following is also known. Theorem F. // K satisfies Frostmaris maximum principle on the domination principle, then it satisfies the continuity principle. VAGUE TOPOLOGY. Let & Q (X) be the totality of continuous real-valued functions with compact support in X. The vague topology is defined on the space of positive measures by the semi-norm \fdμ- The following theorems are very important in the potential theory. Theorem G (BOURBAKI [3]). Let H be a subset of the space of positive measures. If, for every compact set FcX, sup μ ///<&(F)<^ + oo, then H is relatively compact with respect to the vague topology. Theorem H (OHTSUKA [12]). Let a symmetric kernel K satisfy the continuity principle. If a subnet T= {μ ω ω e D, D is a directed set} of a sequence of positive measures, supported by a fixed compact set, converges vaguely to a positive measure μ 0, then lim Kμ ω (x) > Kμ o (x) everywhere in X and lim Kμ t.{x) = Kμ Q (x) nearly everywhere in X.

6 46 M. KISHI CHAPTER I. NECESSARY CONDITIONS 1. Non-degnerate kernels Throughout this paper we are concerned with positive symmetric kernels and we assume that every open set is of positive capacity. By Theorems B' and C we can easily verify that if a kernel K satisfies the V-principle and the equilibrium (resp. balayage) principle, then it satisfies the EU- (resp. BU-) principle. We shall say that a kernel K(x, y) is degenerate if there exist distinct points x x and x 2 such that ^fa'*) == constant in X. K(x 2y x) When K is not degenerate, it is called non-degenerate. With this terminology we give a necessary condition for the unicity principles. Theorem 1.1. In order that a kernel K satisfies one of the U-, BUand ΈXJ-principles, it is necessary that it is non-degenerate. Proof. Suppose that K is degenerate, i.e., there exist distinct points x 1 and x? and a positive constant a such that (2) K(x ly x) = a.k(x 2y x) in X. Then K does not satisfy the U-principle, since the potentials of distinct measures 6 Xl and a 6 X2 are identical in X by (2), where 6 X denotes a point measure at x with total measure 1. We shall show that K does not satisfy the BU- nor EU-principles. Let if be a balayable kernel. We balayage 8 Xl onto the compact set F={x 19 x 2 }. Evidently 6 Xl itself is a balayaged measure. By (2) it is seen that a measure oc 6 X2 is also a balayaged measure. Hence K does not satisfy the BU-principle. Next let K satisfy the equilibrium principle. Then K(x, x t ) = K{x i9 x) <K(x iy x,) in X (ι = l,2) and by (2) K(x ly x) = K(x 2, x) in X. Therefore μ S x. (i = l, 2) are distinct equilibrium measures of F and X) hence K does not satisfy the EU-principle. We shall say that K-potentials separate points when for any distinct points x x and x 2 in X y there exists a potentials Kμ of a positive measure μ with finite energy such that Kμ(x λ )^Kμ{x 2 ). Theorem In order that K-potentials separate points it is necessary and suβcient that for any distinct points x λ and x 2, K(x 19 x)φk(x 2, x) in X.

7 Unicity Principles 47 Proof. Necessity is obvious. We shall show sufficiency. Suppose that ϋγ-potentials do not separate distinct points x x and x 2. Let G be an open set in X Then the continuous function K(x ly x) K(x 2y x) must vanish at some point x' in G, otherwise it contradicts our assumption Kμ(x 1 ) = Kμ(x 2 ) for any positive measure μ with finite energy. Hence there exists a point x r in G such that K(x ly x f ) = K(x 2y x f ). Consequently, taking the base of fundamental neighborhoods of an arbitrarily fixed point x oy we obtain This completes the proof. K(x ly x 0 ) = K(x 2y x 0 ). Corollary 1. // K is non-degenerate, then K-potentials separate points. Corollary 2. In order that a kernel K satisfies one of the U-, BUand EU-principles, it is necessary that K-potentials separate points. REMARK. The converse of Corollary 1 is not valid in general. In 3, it will be shown that the converse is true for composition kernels. 2. Ninomiya's necessary condition Consider the following condition [S], which was obtained by Ninomiya in his thesis [11]. [S] Let x 0 be an arbitrarily fixed point and let λ be a positive measure with compact support S λ which does not contain x 0. If KX(x) <K(x oy x) on S λ, then KX(x)<^K(x oy x) in some neighborhood of x 0. Ninomiya proved that Condition [S] is necessary for the U-principle, and it is sufficient under additional conditions. In this section we shall show that it is necessary for the BU- and EU-principles. First we remark that if K satisfies Condition [S], then it is nondegenerate. In fact, suppose that K is degenerate, that is, there exist distinct points x 1 and x 2 and a positive constant a such that K(x 19 x) = <x-k(x 2y x) in X. Put \ = a 6 X2. Then, on the support of λ, KX(x 2 ) = K(x ly x 2 ) y and at x = x ly KX{x 1 ) = K{x ly x λ ). Therefore Condition [S] is not satisfied. The converse of the above assertion is not valid, in general. For example, let X be a compact space consisting of three points, x x, x 2 and x zy and let a kernel K be given by a matrix

8 48 M. KISHI 1, 1, 1 3 X 2' 2 1, that is, let K(x iy Xj) (i y j=l y 2,3) be given by the element which is in the i-th row and j-th column. Then K is a symmetric non-degenerate kernel, but Condition [S] is not satisfied. It is easily verified that K satisfies the equilibrium principle. In 3, it will be remarked that if a composition kernel K is non-degenerate and satisfies the equilibrium principle or the balayage principle, then it satisfies Condition [S], Now we shall prove the necessity of [S]. Theorem 1.3. In order that a kernel K satisfies the U- or BUprinciple y it is necessary that it satisfies Condition [S]. Proof. What we have to show is that [S] is necessary for the BU-principle, since the necessity of [S] for the U-principle was proved in Ninomiya's thesis [11]. Suppose that K satisfies the BU-principle and Condition [S] is not satisfied. Then there exist a point x 0 and a positive measure λ with compact support S λ which does not contain x 0 such that KX{x) <K(x o,x) KX(x o )>K(x oy x o ). on S λ and Applying Theorem E to the first inequality, we have KX(x)<iK(x 0y x) in X and hence K\(x 0 ) = K(x oy x 0 ). We show that KX(x) = K(x oy x) nearly everywhere on S λ. contrary suppose that a compact set On the F= {xes λ ; KX(x)<K(x 0y x)-8 Qy S 0 y0} is of positive capacity. Then there exists a positive measure u supported by F such that 1^ = 1 and \Kvdv is finite. We balayage this measure v to the compact set {x 0 }, which is of positive capacity since K(x oy x 0 ) is finite, and we obtain a balayaged measure OL S XQ (<x^>0) such that a.k6 XQ {χ) <Kv{χ) in X and

9 Unicity Principles 49 Then <* K(x 0, x 0 ) = cc-k\(x Q ) = Λ = * K6 XQ (χ 0 )-8 0 = a.k(x oy x o )-'δ 9, which is impossible. Thus we have seen that K\(x) = K(x n9 x) nearly everywhere on S λ. Consequently λ is a balayaged measure of 8 XQ to the compact set F' = S λ \j{x Q }. This contradicts our assumption that K satisfies the BU-principle, since 6 XQ itself is a balayaged measure to F'. Theorem 1.4. In order that a kernel K satisfies the ΈU-principle, it is necessary that Condition [S] is satisfied. Proof. Suppose that K satisfies the equilibrium principle and that Condition [S] is not satisfied, that is, there exist a point x 0 and a positive measure λ with compact support S λ^x 0 such that KX(x) < K(x 0, x) on S λ anά KX(x 0 ) > K(x 0, x 0 ). Since K satisfies the equilibrium principle, K(x 0, x)^ck{x Qy x 0 ) at every point x in X> and from Theorem D it follows that ιdλ<l. Therefore and hence K(x 0, Xo)<KX(x o ) = JϋΓUo, x)dx(x)<k(x Q, x Q y \d\<k(x 0y x 0 ) K(x 07 x) = K(x 0, x 0 ) on S λ and \dx = 1. Consequently the measure s x is an equilibrium measure of a K(x Q9 x 0 ) compact set F' = S λ \j {x 0 }. On the other hand we minimize \Kμ{x)dμ(x) among positive measures μ which are supported by S λ and of total measure 1. Since S λ supports λ with finite energy, there exists a minimizing measure μ 0. It is well known (see, for example, Frostman [7]) that the potential Kμ 0 of μ 0 has the following properties : Kμ o (x) > a positive constant a nearly everywhere on S λ and on S μo.

10 50 M. KISHI Then by Frostman's maximum principle (K satisfies the maximum principle by Theorem: B), Kμ Q (.x) = a Kμ o {x) < a nearly, everywhere on S λ and in X We put v o = μ 0. Then v 0 is an equilibrium measure of S λ and f l a \dv^=. Since λ is a competing measure, we have a = ^Kμ o (x)dμ Q (x) < \κx(x)d\(x) < ^K(x oy x)d\{x) = KX(x Q ) = K(x Qy x o ) y and hence 1< K(x oy x 0 ). Therefore we have K^0(x 0 ) = \K{x oy x)dv o (x) = K(x oy x Q )^dv, = K(x oy x o )>l. Consequently Kv Q (x 0 ) = l and v 0 is an equilibrium measure of F'. K does not satisfy the EU-principle. Thus 3. Composition kernels In this section we deal with composition kernels K and investigate equivalent expressions of "non-degeneracy". Let X be a locally compact abelian group and let k(x) be a positive symmetric function defined on X which is finite and continuous at every point #φθ and (0) = lim&(>)< + c>o. We set K(x y y) = k(x-y). The kernels if thus defined are called composition kernels. Note that if k(x) is locally summable with respect to Haar measure μ in X y then every set is of positive capacity. In fact, let G be an open neighborhood of the origin 0, then there exists an open neighborhood G 1 of 0 such that G x C G, G λ is compact and ι_ k(x)dμ(x) is finite. We can take an open neighborhood G 2 of 0 such that for any x y y'm G 2y x y is contained in G 1. Then the potential Kμ\x) of the restriction of μ to G 2 is bounded on G 2, since Kμ(x) = ι_ k(x y)dμ{y)<l \_ k(z)dμ{z) for any JCG G 2. Therefore G 2 supports a positive measure with finite energy and hence G is of positive capacity.

11 Unicity Principles 51 Theorem 1.5. For a composition kernel K y the following three statements are equivalent'. (1) K is non-degenerate, (2) K-potentials separate points, (3) k{x) is not periodic. Proof. The implication (1)=>(2) follows from Theorem 1.2. The implication (2)φ(3) is immediate. We shall prove the implication (3)i^> (1), that is, if a composition kernel K is degenerate, then there exists a point x o φθ such that k{x) = k{x x 0 ) in X. Suppose that there exist distinct points x x and x 2 and a positive constant a such that K(x ly x) = oc K{x 2y x\ namely, k(x 1 x) = oc*k{x 2 x) in X. Then k(x) = kfa-fa-x)) = α *(* 2 -fo-*)) Putting i o =jr r ji: 2 φθ, we have &O) = α : &( * jt 0 ) in X and particularly ) = <* (*<>) and k(x 0 ) = ock(0), and hence <z = l. Therefore k(x) = k(x x 0 ) in X. Corollary. For a composition kernel K, each of the three statements in Theorem 1. 5 is a necessary condition in order that K satisfies one of the U-, BU- and ΈXJ-principles. Now we state another equivalent expression of " non-degeneracy" for a composition kernel satisfying the balayage or equilibrium principle. Theorem 1.6. Let K be a balayable composition kernel. It is nondegenerate if and only if k(0)^>k(x) at every point *φθ. Proof. First we remark that &(0)>&(#) everywhere in X. In fact, let x 0 be an arbitrarily fixed point. We may assume that k(0) is finite. Then the compact set F= {0} is of positive capacity. Sines K satisfies the balayage principle, we can balayage 6 XQ onto F and we obtain a balayaged measure # 0 and namely, a K 0 (0) = KS XQ (0) and a-kε o (x)<k6 XQ (χ) everywhere in X, a k(0) = k(x 0 ) and a k(x) <Zk(x x 0 ) everywhere in X. From these inequalities it is immediately seen that α:<cl and k(0)>k(x Q ). Consequently k(0)~>k(x) everywhere in X. Now we prove our theorem. Suppose that k(0)^>k(x) at every point *φθ. Then k(x) is not periodic and hence it is non-degenerate by Theorem 1. 5.

12 52 M. KISHI Next, suppose that k(0) = k(x 0 ) at some point x 0 =f= 0. We shall prove that k(x) k(x x 0 ) and k(x) is periodic. By the same argument as in the above remark we obtain k(x) <C k{x x 0 ) and k{x) <C k{x + x 0 ) everywhere in X. Into the second inequality we insert x=y x 0 to obtain k(y x o )<ik(y). Consequently k(x) = k(x x 0 ) everywhere in X. Corollary. If a balayable composition kernel K satisfies one of the U-, BU- and EU-principles, then k(0)^>k(x) at every point Theorem Let K be a composition kernel satisfying the equilibrium principle. It is non-degenerate if and only if k(0)^>k(x) at every point xφo. Proof. It is evident that k(0)>k(x) in X, since K satisfies the equilibrium principle. Same as in the proof of the preceding theorem it is sufficient to prove that if K is non-degenerate, then k(0)^>k(x) at every point #φθ. Suppose that there exists a point ^4=0 such that &(0)<&(*i)> hence kφ) = k(x^) by the above remark. We shall show that k{x) is periodic, k(x) = k(x + x 1 ). Without loss of generality we may suppose that &(0) = k(x λ ) = 1 and that at some point x 2, k(x 2 ) = a Consider an equilibrium measure μ= ( 0 + * 2 ) ^ a 1 + a F= {0, x 2 }. Then at every point x y Kμ(x)<Λ. and hence compact set Here we put x = x 1 JΓx 2 and x= x 1 + x 2, and we obtain k(x λ + x 2 ) < k{x 2 ) and k( x 1 + x 2 ) < k(x 2 ). Hence ^ + jt 2 )<(l and we can repeat the above argument for x 1 J Γx 2 instead of x 2 and we obtain k{x 2 ) = k{ - x, + (x, + x 2 )) < & Consequently we have k(x 2 ) = k(x 1 + x 2 ) for any point x 2 such that k(x 2 This equality holds even if k(x 2 ) = l. In fact, if fc^ + ^Xl, then = k( x 1 x 2 ) = k(x 1 x 1 x 2 ) = k( x 2 ) = k(x 2 ) = 1, which is impossible. Thus we have shown that k(x) = k(x + x 1 ) in X, that is, k{x) is periodic and hence K is degenerate by Theorem Corollary. If a composition kernel K satisfies the equilibrium principle

13 Unicity Principles 53 and one of the U-, BU- and Έ\J-principles y then k(o)^>k(x) at every point jtφo. Closing this chapter, we state two remarks. REMARK 1. If a composition kernel K is non-degenerate and satisfies the equilibrium prίnciple y then it satisfies Condition [S]. In fact, if K is non-degenerate, then k(0)^>k(x) at every point xφo by Theorem 1.7. Suppose that λ is a positive measure with compact support S λ which does not contain the origin, and that K\(x)<Ck(x) on S λ. Then by Theorem D, the total measure of λ is <1 and KX(0) = \k(x) dhx) < *(0) ΛdX< k(0). REMARK 2. Analogous statement as above is valid if a composition kernel satisfies the balayage principle. The proof will be given later. CHAPTER II. APPROXIMATION THEOREMS In this chapter we prepare approximation theorems of continuous functions in order to obtain sufficient conditions for the U-, BU- and EU-principles. Let F be a compact space and K(F) be a family consisting of all real-valued finite continuous functions on F. Let be a subfamily of ( (F). We inquire sufficient conditions in order that 35 is dense in K(F) with respect to the uniform convergence topology on F. Here we require, of course, that those sufficient conditions imposed on 35 should be applicable to a family of potentials. The theorem of Weierstrass and Stone is useful in case that kernels satisfy the domination principle. Theorem 2.1. (BOURBAKI [2], p. 53) Suppose that 35 has the following properties: (a) for any functions f λ and f 2 of 3), the functions (/iv/ 2 )(#) = max (/!(*),/;(*)) and (f 1 Af 2 )(x)^min(f 1 (x),f 2 (x)) belong to 35, (b) for any distinct points x 1 and x 2 in F and for any pair of real numbers a λ and a 2y there exists a function f in 35 such that f(χ i ) = a i (i = l,2). Then 35 is dense in ( (F) with respect to the uniform convergence topology on F. In the next chapter we apply this theorem to deduce sufficient conditions for the U- and BU-principles. In case that kernels satisfy the equilibrium principle, we are per-

14 54 M. KISHI mitted to assume that 3) contains constant-functions and we obtain, from the above theorem, Corollary 1. Suppose that 35 is a vector subsp2ce of ( (F) such that (1) 35 contains constant-functions, (2) for any function! f and f 2 of 3), the functions /iv/ 2 and f λ /\f 2 belong to 55, (3) for any distinct points x x and x z in F, there exists a function f in 35 such that f(x 1 )Φf(x 2 ). Then 3) is dense in ( (F) with respect to the uniform convergence topology on F. Proof. What we have to show is that our vector subspace 3) has the property (b) in Theorem 2.1. Let x x and x 2 be distinct points in F and let a x and a 2 be real numbers. By the property (3) there exists a function / 35 such that /(^)+/fe). Since constant-functions belong to 3), the function a 1 )^\}\ f{x 2 )-f{x 1 ) belongs to 3), and g{xi) = ai (/=1,2). Now we denote by K + (F) the family consisting of all non-negative finite continuous functions of F. Then K + (F) is a half vector space, that is, if /,.6( + (F) (ι = l,2), then Λ+/ 2 Ge + (F) and if /e + (F) and α>0, then # / K + (F). Let ^ be a subfamily of ( + (F). Then Corollary 1 can be expressed as follows : Corollary 2. Suppose that ^ is a half vector subspace of such that (Γ) S) 4 " contains positive constant-functions, (2') /or any functions f x and f 2 of % the function f λ f\f 2 belongs to 3) +, (3 r ) /or αwj distinct points x 1 and x 2 in F y there exists a function f in f such that f(x 1 )φf(x 2 ). Put 3) = {/ f=f 1 -f 2 with / G f (ί = 1, 2)}. Then 3) w rf^«5β /«K(F) TOίA respect to the uniform convergence topology on F. Proof. Evidently the vector subspace has the properties (1) and (3) of Corollary 1. We shall verify the property (2). Let Then fi = gi - ^ with g { and h { + (/ = 1, 2). /ia/ 2 - tei-ajλcft-aj = (

15 Unicity Principles 55 and f/\f 2 belongs to, since (g J ι rh 2 )A(g 2 + h 1 ) belongs to + by assumption (2'). By the identity / 1 V/ 2 =-{(-/ 1 )Λ(-/ 2 )}, the upper envelope /iv/ 2 belongs to, too. Consequently, by Corollary 1, is dense in with respect to the uniform convergence topology. Now we replace the conditions in Corollary 2 by weaker ones. Theorem 2.2. Suppose that + is a half vector space of such that (i) if f belongs to + and 0</<l, f*ew 1-/ fe/owgs to +, (ii) /or #w;y function f of +, /fe function /Λl belongs to +, (iii) /or tfwy distinct points x 1 and x 2 in F y there exists a function f in + such that /OOΦ/OO Tfew + 25 ύfews<? ί» S + (F) W/Y/Z respect to the uniform convergence topology on F. This theorem follows from the following three lemmas. Lemma 2.1. Under the same assumptions in Theorem 2. 2, there exists a function f in +, for any given distinct points x λ and x 2 in F, such that /(*i) = 0, f(x*) = l and 0</<l. Proof. By the condition (iii), there exists an /j6 + which takes distinct values at x Λ and x 2. In case that f ι (x ι )^>f 1 {x 2 )> we multiply f λ by a suitable positive number a so that we obtain f 2 = a 'fi of + such that /;(*!) = l>/ 2 (* 2 ). We put / 3 = l-(/ 2 Λl). Then by conditions (ii) and (i), / 3 belongs to + and 0</ 3 <l, f 3 (x 1 ) = 0 and l>/* 3 (^2)>0. Multiplying / 3 by a suitable positive number β, we have a function / 4 = /3 / 3 of + such that / 4 (^) = 0 and / 4 (JC 2 ) = 1. Then /=/ 4 Λ1 is a function of + and 0<f<l, /(^) = 0 and /(* 2 ) = l. In case that f 1 (x 1 )<^f 1 (x 2 )> w e obtain, by the above argument, a function of + such that 0< <l, g(x 1 ) = l and ^T(Λ: 2 ) = 0. Consequently f=l g is a required function. Lemma 2.2. Suppose that a half vector space + /zαs the properties in Theorem z^ /or <z/iy c/o^ί/ subset A of F and for any point x 0 in the complement of A y there exists a function ge^ such that 0^ "ίίl> g( χ o) = ~L and g(x) = 0 everywhere in A. Proof. Let y be an arbitrarily fixed point of A. Then by Lemma 2.1, there exists a function f y of + such that 0</ y <l, f y (x 0 ) = 0 and f y (y) = l. Since /y is continuous, there is a neighborhood U(y) of jv such that f y^>l/2 everywhere in U(y). Then, A being compact, a finite family { /(.?, )}, 1<^<^, covers A We put

16 56 M. KISHI By the condition (ii), h belongs to 2) + and 0<*<l, h(x o ) = O and h(x) = l everywhere in A. Consequently g=l-h is a required function of 2> +. Lemma 2.3. Suppose that a half vector space S + has the properties in Theorem 2.2. Then for any disjoint closed subsets A and B> there exists an he^+ such that 0<h<l, h = l on A and h = 0 on B. Proof. Let y be an arbitrarily fixed point in A. Then by Lemma 2.2, there exists g y e1s) + such that 0<g y <l, g y (y) = l and g y =0 on B. By the continuity of g y9 there is a neighborhood U(y) of y, at every point of which g y^>l/2. We can cover i by a finite family of these neighborhoods : 0 U(y 4 ) 5 A. Put g=σg,. Then g belongs to + and 1 = 1 1 = 1 * it vanishes on B and it is greater than 1/2 on A Consequently h=2gal satisfies the conditions required. Now we prove Theorem 2.2. Let φ be a positive finite continuous function on F. We prove that a sequence {/ } of functions of 2) + converges uniformly to φ. Without loss of generality we may suppose that 0< ><l. We put for each *, l<k<n 9 A k = {x; φ(x)>k/n}, B k = {x; ψ(x)<(k-l)/n). Then A k and B k are disjoint closed sets. By Lemma 2. 3, there exists a function g* 25 + such that 0<^<l, g k = l on ^ and g k = 0 on β Λ. Putting we have \φ f n \<\ln everywhere on F. Consequently {/J converges uniformly on F to φ. CHAPTER III. SUFFICIENT CONDITIONS 1. Lower envelope principles DEFINITION. We say that a kernel K satisfies the strong lower envelope principle, when, for any positive measures μ and v, one of which has finite energy, there exists a positive measure λ such that KX = Kμ Λ Kv nearly everywhere in X. DEFINITION. We say that a kernel K satisfies the compact lower envelope principle, when, for any compact set FcX and for any positive

17 Unicity Principles 57 measures μ and v, one of which has finite energy, the lower envelope Kμ/\Kv coincides nearly everywhere on F with a potential KX of a positive measure λ supported by F. DEFINITION. We say that a kernel K satisfies the lower envelope principle, when, for any compact set F and for any signed measure <r=μ 1 μ 2 with μ (i=l,2) of positive measures with finite energy, the lower envelope Kσ Λ 1 coincides nearly everywhere on F with a potential Kr of a signed measure τ = v 1 v 2y where P (i=l,2) are positive measures with finite energy. The notion of the lower envelope principles was first introduced by Deny [6]. He proved the equivalence of the strong lower envelope principle and the domination principle for his distribution-kernels of positive type. The principle was also investigated by Choquet-Deny [4] for the kernels defined on a finite space under the name of the lower envelope principle. For later use we investigate in this section the relations among the above principles and the domination principle. First we state Lemma 3.1. Let K be a kernel of positive type, and let μ λ and μ 2 be positive measures whose energy are finite. If the energy of μ^ μ 2 vanishes, i.e., 11 μ x μ = I Kμ 1 dμ 1 2 I Kμ x dμ 2 + I Kμ 2 dμ 2 = 0, then Kμ x = Kμ 2 nearly everywhere in X. Proof. υ On the contrary suppose that there exists a compact set F of positive capacity on which Kμ^^Kμ^x) and \\μ 1 μ 2 \\ 2 = 0 The compact set F supports a positive measure λ with finite energy, by which we integrate Kμ 1 Kμ 2 and, noting that K is of positive type, we obtain which is a contradiction. By this lemma we have Theorem 3.1. If a kernel of positive type satisfies the compact lower envelope principle, then it satisfies the domination principle. Proof. Suppose that K is of positive type and satisfies the compact lower envelope principle. We shall show that K satisfies the domination principle. Let x 0 be an arbitrarily fixed point and μ be a positive measure 1) This proof is due to Ohtsuka.

18 58 M. KISHI with compact support S μ which does not contain x 0 and let Kμ{x)<CK(x 0, x) on S μ. By Theorem E, it is sufficient to show that the same inequality holds everywhere in X. On the contrary suppose that there exists a point x 1 such that KμixJ^KiXo, x λ ). Then by the lower semi-continuity of Kμ, there exists a neighborhood U of x 1 with compact closure such that S μ r\u=φ and (3) Kμ{x)>K(x 09 x) in U. By the compact lower envelope principle, Kμ{x) Λ KS XQ (X) coincides nearly everywhere on F=S μ.\ju with a potential Kv of a positive measure v supported by F. (4 ) Kv(x) = Kμ{x) A K(x 0, x) nearly everywhere on F. Then, noting that K is of positive type and that μ and v have finite energy, we obtain -p)=[ [ {Kμ-Kv)dμ-[ {Kμ-Kv)dv = -( (Kμ-Kv)dv<0, J F namely the energy of the signed measure μ v vanishes. by Lemma 3.1, Consequently, Kμ{x) = Kv{x) nearly everywhere in X. Therefore by (4), Kμ{x)<ZK(x 0, x) nearly everywhere in U. This contradicts (3), since every open set is of positive capacity. This completes the proof. REMARK. There exists a kernel K which satisfies the strong or compact lower envelope principle but not the domination principle. This kernel is necessarily not of positive type. We quote the following example from [4], p. 131: let X be a finite space consisting of three points x ly x 2 and x 3 and let K be given by the matrix This kernel satisfies the strong lower envelope principle but not the domination principle. Now suppose that K satisfies the domination principle. We shall

19 Unicity Principles 59 examine whether K satisfies the strppg lower envelope principle. The following is rather obvious. Lemma 3.2. // Kμ x and Kμ 2 are finite-valued continuous potentials, then for any compact F, the lower envelope Kμ 1 /\Kμ 2 coincides nearly everywhere on F with a potential KX of positive measure λ supported by F. Proof. We note that K satisfies the continuity principle, since it satisfies the domination principle (Theorem F). Put Then / is continuous in X. By Gauss' (or Ninomiya's) variational method, the existence of the following positive measure λ is shown: (i) λ is supported by F. (ii) KX(x) >f(x) nearly everywhere on F, (iii) KX(x)<f(x) on S λ. We shall show that KX(x) =f(x) nearly everywhere on F. By (ii) K\(x)<^Kμi(x) on S λ (ί = l,2). Hence λ has finite energy and, by the domination principle, K\{x)< 1 Kμ i {x) in X (i = l, 2), and hence KX{x)<Lf(x) in X. Together with (ii), we obtain KX(x) =f(x) nearly everywhere on F. Thus the proof is completed. By the same method we can prove Lemma Let Kμ 1 be a continuous potential. Then for any potential Kμ 2 of a positive measure μ 2 and for any compact set F, the lower envelope Kμ λ Λ Kμ 2 coincides nearly everywhere on F with potential KX of a positive measure λ supported by F. Proof. Put f=kμ 1 ΛKμ 2. For an arbitrary positive number N, put K N μ 2 (x) = \ j K N (x,y)dμ 2 (y) f N (x) = KμXx) Λ K N μ 2 (x). Then K N μ 2 is continuous in X y and K N μ 2 (x) f Kμ 2 {x) and f N ( χ )\f( χ ) (as N\ oo) at every point x X. By Gauss' (or Ninomiya's) variational method there exists a positive measure λ^ such that (i) λ^ is supported by F, (ii) KX N 7>f N nearly everywhere on F, (iii) KX N <Zf N on S\ N. By (iii) and the domination principle we have (iiiy K\ N <f in X

20 60 M. KISHI First we note that the tσΐal measures id\ N since N {)<±f N (χ) < * (xe S, N ), (iv=l, 2, ) are bounded, where cc=mf FXF K(x y j)>0 and M=max F Kμ 1 (x). Therefore by Theorem G a subnet T= {λ ω &e D} of the sequence {λ^} converges vaguely to a positive measure λ supported by F. Then by Theorem H KX(x) < lim KX ω (x) everywhere in X and ω Kλ(x) = lim K\(x) nearly everywhere in X. ω Hence by (iii)', KX(x)<f(x) everywhere in I and by (ii), KX(x)>f(x) nearly everywhere on F. Consequently KX(x)=f(x) nearly everywhere on F. This completes the proof. The following lemma was proved in [9]. Lemma 3.4. If K satisfies the continuity principle, then for any positive measure μ with finite energy, there exists a sequence {μ n } of positive measures with the following properties: 1 {μ n } converges vaguely to μ y 2 the potentials Kμ n are all finite-valued continous in X, and 3 Kμ n {x) Kμ(x) at every point x of X. We call each of the sequence {μ n } a smoothed measure of μ. Now we prove Theorem 3.2. If K satisfies the domination principle, then it satisfies the compact lower envelope principle. Proof. Let F be a compact set, and let /* be a positive measure with finite energy and v be a positive measure. We shall show that the lower envelope f=kμf\kμ coincides nearly everywhere on F with a potential KX of a positive measure λ supported by F. By Lemma 3. 4, a sequence {μ n } of smoothed measures of μ converges vaguely to μ. Put f n = Kμ Λ ΛKi>. Then by Lemma 3. 3, each / coincides nearly everywhere on F with a potential KX n of a positive measure X n supported by F; f n = KX n nearly everywhere on F. The total measures \dx n («= 1, 2, ) are bounded, since

21 Unicity Principles 61 '< JJϋΓ(*, y)d\ n dλ n = where a = inf FxF K(x, y) > 0. Hence by Theorem G, a subnet T = {λ ω ; ω θ} converges vaguely to λ. Then id^ <lim If λ ω in X and /Γλ=lim KX ω nearly everywhere in X Therefore KX = lim f n = f nearly everywhere on F. Consequently K satisfies the compact lower envelope principle. Here we give an example of a kernel which satisfies the compact lower envelope principle, but not the strong lower envelope principle. Let X be a space consisting of a countably infinite number of points^ {x 17 x 2> }, and let a kernel K be given by K{x ifxi ) = 2 (ι = l,2,.. ) K(x i9 Xj) = 1 (/ 4= h i, 1 = 1,2,-). This symmetric kernel K satisfies the domination principle and hence the local lower envelope principle by Theorem 3.2. In fact, let μ be a positive measure supported by F= {x nχj x n2,, x n } 9 μ= Σl^i 6x n (m,0>0), and let x' be a point F and Kμ{x nt )<^K(x niy x') for every ι\ Then Therefore for any x F> Kμ(x) = Σ m s < 1 Consequently by Theorem F, K satisfies the domination principle. Now let μi = xi (/ = 1,2). Then the lower envelope f=kμ 1 ΛKμ 2 is constantly 1 in X and does not coincide in X with a potential of a positive measure supported by a compact set, since the function 1 is not a potential. Thus K does not satisfy the strong lower envelope principle. By this example it is also shown that a kernel does not necessarily satisfy the strong lower envelope principle even if it satisfies the domination principle or the balayage principle. Thus the following problem arises: Find a sufficient condition in order that a balayable kernel satisfies the strong lower envelope principle.

22 62 M. KISHI Here we give a sufficient condition, which is not necessary as shown later. Let K be of positive type and denote by Θ + the totality of positive measures with finite energy. The semi-norm IIA*-»II = defines the strong topology on +. When any strong Cauchy net in ( + converges to an element + with respect to the strong + is, by definition, strongly complete. When K satisfies the balayage principle, it is of positive type by Theorem A and we can prove Theorem 3.3. Suppose that K satisfies the balayage principle and that + is strongly complete. Then K satisfies the strong lower envelope principle. Proof. Let μ and v be positive measures with compact support and μ & +. We shall show that f=kμ/\kv coincides nearly everywhere in X with a potential KX of a positive measure. Take a compact set F o which contains S μ \js^, and denote by D the directed set of compact sets F containing F o. To each F of D a positive measure X F supported by F corresponds, by Theorem 3. 2, in such a way: KX F = / nearly everywhere on F. Put T={X F ; FeD}. Then T is a strong Cauchy net, since for any FcF', KX F <ikx F ' nearly everywhere in X and 0 < \κx F dx F -2 Thus, by the strong completeness of Gf +, T converges strongly to a positive measure λ e 6? +. As is easily seen = lim [κx F dτ for any T +. and hence KX = limkx F nearly everywhere in X. Consequently KX=f nearly everywhere in X. As to the strong completeness of + we refer to Fuglede [8] and Ohtsuka [12]. They gave sufficient conditions for the strong completeness. The following example is contained in [12]. Exclude the closed unit ball with center at the origin from the 3-dimensional Euclidean

23 Unicity Principles 63 space and take the rest for X, and consider the Newtonian kernel K in X Then K satisfies the domination principle and (S + is not strongly complete. But as easily seen, it satisfies the strong lower envelope principle. Now we state relations among the lower envelope principle and the equilibrium principle. First we prove Theorem 3.4. Suppose that K is of positive type and that if Kμ is the potential of a positive measure with finite energy, then the lower envelope Kμ Λ1 coincides nearly everywhere on a given compact set with the potential Kv of a positive measure supported by the compact set. Then K satisfies the equilibrium principle and Frostmarts maximum principle. Proof. We shall show that K satisfies Frostman's maximum principle. Let /i be a positive measure supported by a compact, set F and Kμ{x)<A. on F. If there exists a point x^f such that Kμ{x^>\ y then the same inequality holds in a neighborhood U of x x with compact closure such that Fr\ϋ=0. By our assumption the lower envelope f(x) = Kμ{x) A1 coincides nearly everywhere on F' = F\JU with a potential Kv(x) of a positive measure supported by F'. This leads to a contradiction as in the proof of Theorem 3.1. The converse of Theorem 3. 4 is not true there exists a kernel K which satisfies the equilibrium principle but the lower envelope KμAl is not a potential. Let X be a compact space consisting of four points, x lf x 2y x 3 and x 4, and let a kernel K be given by a matrix 3 2' 1 2' 1, 1, 1 2' 3 2' 1, 1, As easily seen, this kernel satisfies the equilibrium principle, but the lower envelope KμAl, μ=s x^y is not a potential, since the determinant of the matrix vanishes. This kernel does not satisfy the lower envelope principle. We shall come back to this example in 4. Theorem Let K be a kernel satisfying the equilibrium principle. If it satisfies the compact lower envelope principle or the domination principle, then it satisfies the lower envelope principle. 1, 1, 3 2' 1 2' )

24 64 M. KISHI Proof. By Theorem 3.2, it is sufficient to prove that if K satisfies the equilibrium and compact lower envelope principles, then it satisfies the lower envelope principle. Let F be a compact set and <r=μ 1 μ 2 be a signed measure such that μ g (/=1,2) are positive measure with finite energy and supported by F. Then KσΛl = Kμ 1 Λ(l+Kμ 2 )-Kμ 2 and 1 coincides with an equilibrium potential KX of F nearly everywhere on F. Hence, by the compact lower envelope principle, Kμ λ /\ (1 + Kμ 2 ) coincides nearly everywhere on F with a potential Kv of a positive measure supported by F. Consequently K satisfies the lower envelope principle. Note that if σ is a positive measure, then Kr = Kσ Λ 1 is a potential of a positive measure. Naturally there exists a kernel which satisfies the equilibrium and lower envelope principles but not the compact lower envelope principle. For example, let X= {x 1, x 2, x 3 } and K be given by As easily seen, K satisfies the equilibrium principle but not the domination principle. Hence by Theorem 3.1, it does not satisfy the compact lower envelope principle. Since the determinant of the matrix does not vanish, K satisfies the lower envelope principle. Closing this section we state the relations among three envelope principles in the following diagram: CD strong lower env. prin. ^~T > compact lower env. prin. (2) (3) \i (4) (5) ( (6) lower env. prin. The relations (1), (2) and (6) have been already shown (3) is shown by a kernel on a compact space, which satisfies the domination principle but not the equilibrium principle from this and (1) follows (5) (4) is shown by a kernel given to show (2). 2. Regular kernels In the potential theory "regularity" is the important notion, from which it follows that if a potential is continuous on a compact set F y its balayaged potential to F is continuous.

25 Unicity Principles 65 DEFINITION. We shall say that a compact set F is regular with respect to a kernel K or /f-regular when, for any positive continuous function h(x) on F and for any potential Kμ{x) of a positive measure μ y if Kμ(x)'>h(x) nearly everywhere on F, then the same inequality holds everywhere on F. DEFINITION. We shall say that a kernel K is regular 0 when, for any compact set F and for any open neighborhood G of F, there exists a if-regular compact set F' such that FcF' CG. From the classical potential theory it is seen that the kernels of order a on the Euclidean spaces and the Green kernels on open Riemann spaces are regular. Evidently a kernel K is regular if K{x 9 x) is finite at every point x. The following is essentially contained in Frostman Theorem Let X be the m-dimensional Euclidean space. In order that a kernel K is regular, it is sufficient that it satisfies the condition : For any point x 0 6 X, there exist an open neighborhood U of x 0 and a positive constant A = A(x 0 ) such that \ I K(x, y)dxdy is finite, J UJ U and for any positive measure μ whose support S μ is contained in U and for any ball B(x oy r) in U, with center at x 0 and radius r, ( Kμ{x)dx<A Kμ(x 0 ), where v r denotes the volume of B(x 0) r). Proof. Let us assume our condition. We first show that a compact set F is ZΓ-regular, if it satisfies the condition of Poincare, that is, for every point x of F, there exists in F a cone with vertex at x. Let h(x) be a positive continuous function on F and let Kμ(x) be a potential of a positive measure. Suppose that ( 5 ) Kμ{x) > h(x) nearly everywhere on F. What we have to show is the validity of Kμ{x)>h(x) everywhere on F. Let x Q be an arbitrarily fixed point of F> and let c be a cone in F with vertex at x 0. We denote by c r the intersection cr\b(x Q9 r), which is contained in Ur\F for every sufficiently small r, and by u r the volume of c r. Then u r /v r =a<^l. Since we may suppose that Kμ{x 0 ) is finite, we can choose, for any positive number, a positive number r x such that 2) In the sense of Choquet regular kernels are those which satisfy the continuity principle.

26 66 M. KISHI (6) g where μl is the restriction of μ to B(x Qy r x ). Because of the continuity of h(x) and Kμ"(x), μ" = μ μ, there exists a positive number r 2 <jr x such that (7) *(*<>)< *(*) + -f- in B(x o,r 2 )r\f, (8) #/'(*) <#/'(*o)+j- in B(x o,r 2 ). By (7) and (5), we have 3 «1 Γ 6 Here the second term I Kμ\x)dx is not greater than Kμ"(x o ) + u r2 i e r 2 3 by (8), and the first term ( Kμ\x)dx = -ί- ( Kμ\x)dx /( Λ ) dx < A-Kμ(x 0 ) < A, V r2 BQx o,r 2 ) a 3 by the conditions in our theorem and (6). Thus we obtain h(x 0 ) < Kμ"{x Q ) + S Consequently h{x Q )<ZKμ{x Q ) y and hence F is iγ-regular. Now we show that for any compact set F and for any open neighborhood G of F, there exists a if-regular compact set F' such that FcF' CG. For this purpose we take a compact set F' which is enclosed by surfaces, pararell to the coordinate-axes. This F r is if-regular, since it satisfies the condition of Poincare. This completes the proof of the theorem. Corollary. The oc-kernels y 0<^cc<^m y on the m-dimensional Euclidean space are regular. Now we state some consequences of "regularity". Theorem Let K be balayable. If a potential Kμ of a positive

27 68 M. KISHI Kμ(x) of positive measures with finite energy, and let (F) be a totality of continuous functions f=f 1 f 2 with / Z + (F) (/ = 1,2). Lemma Let K be regular and balayάble y and let F be K-regular. Then (F) is closed with respect to the operations v and Λ. Proof. It is sufficient to show that (F) is closed with respect to the operation Λ, since fvg= - {( /)Λ( g)}. Let/and g be in (F). Then f=f,-f 2y g=g 1 -g 2 with /, and in gl + (F) (/-1,2), and Hence, + (F) being a half vector space, it is sufficient to show that if h γ and h 2 are in + (F), then h x ι\h 2 is also in + (F). Write hi = /ϋ> f on F (/*,- > 0). Then μ,i have finite energy and, by Theorem 3. 9, h x A h 2 coincides on F with a potential KX of a positive measure supported by F, and hence A X AA 2 belongs to + (F). Lemma Let K be regular and balayable, and let F be a compact set. If K is non-degenerate, then for any distinct points x 1 and x 2 of F and for any pair of real numbers a 1 and a 2y there exists a function f in (F) such that f(x i ) = a i (ί = l,2). Proof. We first show that there exist positive measures μ x and μ 2 with finite energy such that Kμj(x) (j = l>2) are continuous in X and the determinant ) K μi (x 2 ) Kμ 2 (x λ ) Kμ 2 (x 2 ) Since K is non-degenerate, there exist distinct points y s (j = l, 2) such that y 1 ),K(x ly y 2 ) φ K{x 2y y λ ) K(x ly y,)' When K(y Jy yj) (j = l or 2) is finite, we put μ~ y r Then μj is a positive measure with finite energy and Kμj(x) is continuous and Kμ s (x) = K(x y yj) in X When K(y Jy yj) (j = l or 2) is infinite, we take a neighborhood Gj of y jy such that XifcGj (ί = l, 2). Then the compact set F Gj contains x (i = l y 2) but not y 5. By Theorem 3.8 there exists a positive measure μ s with finite energy such that Kμj is continuous in X and Kμj(x) = K(x y y/) everywhere on F Gj. Consequently Kμ = K(x iy yj) (7 = 1,2). Thus we have constructed positive measures

28 Unicity Principles 67 measure μ is continuous on a K-regular compact set F r, then a balayaged potential Kμ of Kμ onto F f is continuous on F'. Theorem 3. T. Let K satisfy the equilibrium principle. If a compact set F / is K-regular, then the equilibrium potential of F / is continuous on F'. These two theorems follow immediately from the definition of " regularity ". Theorem Let K be regular and balayable y and let x 0 be a point which is not contained in a compact set F. Then there exists a positive measure μ with finite energy such that Kμ(x) is continuous in X and Kμ{x) = K(x, x 0 ) everywhere on F. Proof. Let G be an open neighborhood of F which does not contain x 0. The kernel K being regular, there exists a Zf-regular compact set F' such that FcF'cG. Let μ be a balayaged measure of S XQ onto F'. Then by Theorem 3. 7, Kμ{x) is continuous on F f 5S μ and Kμ{x) = K(x, x 0 ) everywhere on F\ Since K satisfies the continuity principle by Theorem F, Kμ{x) is continuous in X. Theorem Let K be regular and balayable y and let F' be a Irregular compact set. If μ and v are positive measures, one of which has finite energy, and if Kμ{x) and Kv(x) are continuous on F r, then the lower envelope Kμ{x) Λ Kv(x) coincides everywhere on F' with a potential KX{x) of a positive measure λ supported by F\ Proof. Put f(x) = Kμ{x) ΛKV(X). Then f(x) is continuous on F'. Since K is balayable and hence it satisfies the domination principle, it satisfies the compact lower envelope principle by Theorem Hence there exists a positive measure λ supported by F' such that KX{x) = f(x) Again by the domination principle, nearly everywhere on F'. KX(x)<ίf(x) everywhere in X. Then by the K-regularity of F f, it is shown that KX(x) = f(x) everywhere on F'. 3. Sufficient conditions for the U- and BU-principles In this section we give sufficient conditions in order that balayable kernels satisfy the U- and BU-principles. Let F be a compact set, and let + (F) be a totality of non-negative finite continuous functions f(x) on F which coincide on F with potentials

29 Unicity Principles 69 (.7 = 1,2) with finite energy such that Kμj(x) are continuous in X and the determinant Kμfa) Kμ λ (x 2 ) ^ Q Kμ 2 (x 1 ) Kμ 2 (x 2 ) Now, for any real numbers ξ 1 and ξ 29 ξ 1 Kμ 1 (x) + ξ 2 Kμ 2 (x) belongs to the family Φ(F) and, since the determinant above does not vanish, there exists a function f(x) = ξ 1 Kμ 1 (x) J rξ 2 Kμ 2 (x) of 55(F) such that h(x i ) = a i (ί = l,2). Now we can prove an approximation theorem. Theorem Let K be regular, balayable and non-degenerate, and let F be a K-regular compact set. Then 3)(F) is dense in &(F) with respect to the uniform convergence topology. Proof. By Lemmas 3.5 and 3.6, (F) has the properties (a) and (b) of Theorem 2.1. Hence our theorem follows from Theorem 2.1. By this theorem we obtain a sufficient condition for the U- and BU-principles. Theorem Let K be regular and balayable. In order that it satisfies the U-principle y it is sufficient that it is non-degenerate. Proof. Let μ 1 and μ 2 be positive measures supported by a compact set F, and Kμ ± = Kμ 2 nearly everywhere in X. We shall show the equality μ 1 = μ 2. Take a if-regular compact set F f containing F. Then by Theorem 3.10, (F') is dense in (F 7 ) with respect to the uniform convergence topology on F'. Hence, for any finite continuous function / and for any positive number, there exists a function h = Kv x Kv 2 of (F') such that and hence \Λx)~h(x)\< on F', \(f(x)-h(x))d μi (x) I < S ^d μi < S' (ί = 1, 2), where ' tends to 0 with 6. Since v i (Z = l,2) have finite energy, \ Kμ x d»i = I Kμ 2 d»i. Consequently \h{x)dμix) = = \Kμ 2 dv 1 \Kμ 2 dv

30 70 M. KISHI v χ dμ 2 \Kv 2 dμ 2 = γι(x)dμ 2 (x). Therefore \\f(x)dμ 1 (x)-[f(x)dμ 2 (x)<^2s' and hence [f(x)dμ 1 (x) = \f(x)dμ 2 (x) for any finite-valued continuous function / on F'. Thus μ 1 = μ 2. This completes the proof of the theorem. Corollary 1. If a regular kernel K satisfies the BΛ3-principle, then it satisfies the ^-principle and the energy principle. Proof. If a regular kernel K satisfies the BU-principle, then, by Theorem 1.1, it is non-degenerate and by Theorem 3.11, it satisfies the U-principle, and hence the energy principle. Corollary 2. Let K be regular and balayable. In order that it satisfies the BXJ-principle, it is sufficient that it is non-degenerate. Proof. This follows immediately from Theorem 3.11, since the U-principle implies the BU-principle. The following was proved by Ninomiya [11]. Theorem Let K be regular and balayable. In order that it satisfies the U- and BU-principles, it is sufficient that is satisfies Ninomiya's condition [S] (in Chap. I, 3). Proof. This follows from the remark in 3 of Chap. I if K satisfies Condition [S], it is non-degenerate. In case that K satisfies both the balayage and equilibrium principles, a weaker condition than non-degeneracy is sufficient for the U- and BU-principles. Theorem Let K be regular and satisfy both the balayage and equilibrium principles. In order that it satisfies the U- and BXJ-principles y it is sufficient that K-potentials separate points. Proof. It is sufficient to prove the sufficiency for the U-principle. Let μ 1 and μ 2 be positive measures with compact support such that Kμ λ = Kμ 2 nearly everywhere in X. By the regularity of K we may suppose that they are supported by a if-regular compact set F. Evidently the half vector space + (F) prossesses the property (Γ) in Corollary 2 of Theorem 2.1. By Lemma 3. 4, it possesses (2') of the corollary. We verify the property (3 7 ). By our assumption there exists a potential Kv of a positive measure with finite energy which separates distinct points

31 Unicity Principles 71 x λ and x 2, and by Lemma 3.3, there exists a smoothed measure 1/ of» such that Kv'(x λ )^Kv'(χ 2 ). Consequently by the corollary, (F) in dense in K(F) with respect to the uniform convergence topology. Then we obtain the equality μ ± = μ 2 by the same way as in the proof of Theorem REMARK 1. If a regular kernel satisfies only the balayage principle, the condition "if-potentials separate points" is not sufficient for the U- nor BU-principleβ. In fact, if X= {x,, x 2 } and K is given by 4, 2 2, 1 then K is regular and balayable, and /f-potentials separate points, but K does not satisfy the U- nor BU-principles. REMARK 2. In case that K is not regular, the author does not know whether the statements in Theorems 3.11 and 3.12 are true. We can prove only that if K satisfies both the balayage and equilibrium principles and K-potentials separate points, then it satisfies the following restricted unicity principle {and hence the analogous restricted balayage-unicity principle) : if μ x and μ 2 are positive measures with finite energy such that Kμ λ = Kμ 2 nearly everywhere in X, then μ 1 = μ 2. This is verified as follows. Denote by ί(f) the totality of non-negative finite continuous functions / which coincide nearly everywhere on a compact set F with potentials Kμ of positive measures with finite energy, and put Evidently S) + (F)C o + (/ Γ ) and (F)C 0 (^) By our assumptions 0 (F> has the properties (1'), (20 and (3 r ) of Corollary 2 of Theorem 2.1. Hence 0 (^) is dense in ( (F) with respect to the uniform convergence topology. Consequently for any finite continuous function / on F by which μ x and μ 2 are supported and for any positive number, there exists a function h in 0 (^) such that on F and h = Kv 1 Kv 2 nearly everywhere on F. Since μ 4 and v i (/=1,2) have finite energy, we obtain Hence μi = μ 2. \hdμ 1 = \hdμ 2 and \fdμ ι = \fdμ 2.

32 72 M. KISHI Now we state sufficient conditions in order that a composition kernel satisfies U- and BU-principles. Theorem Let K be a regular composition kernel satisfying the domination principle. Each of the following conditions is sufficient for the U- and BU-principles: (1) K is non-degenerate y (2) K-potentials separate points, (3) k(x) is not periodic, (4) k(0)>k(x) at every point xφo, (5) K satisfies Ninomiya's condition [S] (in Chap. I, 2). Proof. This follows immediately from Theorems 1. 5, 1. 6 and 3.11 and the remark in 2 of Chapter I. Choquet-Deny [5] announced the condition (3) without proof. The condition (4) was obtained by Ninomiya [11] assuming both the balayage and equilibrium principles. The sufficiency of (4) was proved by the author assuming only the balayage principle [10]. Here we prove the statement in Remark 2 in Chapter I: if a balayable composition kernel is non-degenerate, it satisfies Condition [S]. We may suppose that k{ϋ) is finite, since otherwise it satisfies Condition [S]. Then K is regular. Hence by Theorem 3.14 it satisfies the BU-principle, and by Theorem 1. 3 it satisfies Condition [S]. Summing up the results obtained in Chapters I and III, we state Theorem. Let K be a symmetric regular balayable kernel. In order that K satisfies the U- or BU-principle, it is necessary and sufficient that K is non-degenerate. It is also necessary and sufficient that K satisfies Ninomiya y s condition [S]. Theorem. Let K be a symmetric regular kernel satisfying both the balayage and equilibrium principles. In order that K satisfies the U- or BU-principle, it is necessary and sufficient that K-potentials separate points. Theorem. Let K be a symmetric regular balayable composition kernel. Each of the five conditions in Theorem 3.14 is necessary and sufficient for the U- or BU-principle. 4. Sufficient condition for the U- and EU-principles In this section we assume that K satisfies the equilibrium principle and we give a sufficient condition in order that K satisfies the U- and EU-principles. First we remark that any condition considered in the preceding section is not sufficient. In fact, the kernel given soon after Theorem 3.4 does

33 74 M. KISHI by Theorem C, and Therefore I fdμ 1 = \ fdμ 2, a nd hence μ λ = Corollary. Let K satisfy the equilibrium principle and the lower envelope principle, and let K-potentials separate points. Then K satisfies the energy principle and the restricted unicity principle (in Remark 2 to Theorem 3.14). REMARK. In 3, we have shown that the following implication is valid for regular kernels: the BU-principle =#> the U-principle. The analogous implication, the EU-principle ==> the U-principle, seems to be false, but the author has not yet any example which shows the EUprinciple =j=> the U-principle. Mathematical Institute, Nagoya University (Received October 10, 1960) References [ 1 ] G. Anger: Sur le role des potentiels continues dans les fondements de la theorie du potentiel, Seminaire de theorie du potentiel, 2 e annee (1958), 30 pp. [ 2 ] N. Bourbaki: Topologie generate. Chap. X, Paris [ 3 ] N. Bourbaki: Integration, Paris [ 4 ] G. Choquet and J. Deny: Modeles finis en theorie du potentiel. J. d'analyse Math., 5 (1956/57), [ 5 ] G. Choquet and J. Deny: Aspects lineaires de la theorie du potentiel. Theoreme de dualite et applications, C. R. Acad. Sci. Paris, 243 (1956), [ 6] J. Deny: Les potentiels d'energie finie, Acta Math., 82 (1950), [7] O. Frostman: Potentiel d'equilibre et capacite des ensembles. These, Lund (1935), [ 8 ] B. Fuglede: On the theory of potentials in locally compact spaces, Acta Math., 103 (1960), [ 9 ] M. Kishi: On a theorem of Ugaheri, Proc. Japan Acad., 32 (1956), [10] M. Kishi: On a topological property of a family of potentials, Japanese J. Math., 29 (1959), [11] N. Ninomiya : Etude sur la theorie du potentiel pris par rapport au noyau symetrique, J. Inst. Poly. Osaka City Univ., 8 (1957), [12] M. Ohtsuka: On potentials in locally compact spaces, J. Sci. Hiroshima Univ., Ser. A, I (to appear).

34 Unicity Principles 73 not satisfy the U- nor EU-principle, since the determinant of the matrix vanishes, but it is non-degenerate and it satisfies Ninomiya's Conditions [S] and potentials separate points. This remark is an answer to the following Ninomiya's problem, raised in his thesis: Is it true that Condition [S] is sufficient in order that a symmetric kernel of positive type satisfies the U-principle? By the above remark the answer is negative. Thus we need to give an additional condition to [S], which should be weaker than the balayage principle, since, by Theorem 3.13, a regular kernel satisfies the U- principle if it satisfies Condition [S] and the balayage and equilibrium principles it is also seen that it satisfies the EU-principle. Denote by {F) the family consisting of all non-negative finite-valued continuous functions on a compact set F which coincide nearly everywhere on F with potentials K<τ of signed measures σ=μ v such that μ and v are positive measures with finite energy. Theorem If K satisfies the equilibrium principle and the lower envelope principle and if K-potentials separate points, then, for any compact set F> (F) is dense in & + (F) with respect to the uniform convergence topology. Proof. It is sufficient to verify that {F) has the properties (i), (ii) and (iii) of Theorem 2.2. The properties (i) and (ii) are immediate. We shall show that (F) has the property (iii). Since if-potentials separate points, there exists, for any given distinct points x 1 and x 2 of F, a potential Kμ{x) of a positive measure with finite energy such that Kμ{x^)φKμ{x 2 ). Then by Lemma 3.3, there exists a smoothed measure μ of μ such that Kμ is finite continuous in X and Kμ{x^)φKμ{x 2 ). Thus {F) has the property (iii). Consequently, by Theorem 2.2, (F) is dense in ( Theorem Let K satisfy the equilibrium principle and the lower envelope principle. In order that K satisfies the ΈλJ-principle it is sufficient that K-potentials separate points. Proof. Let μ x and μ 2 be equilibrium measures of a compact set F, and let / be an arbitrary finite continuous function on F. Then by Theorem 3.15, for any positive number, there exists a function he (F) such that h = Kv ί -Kv 2 and /-* < on F. Since μ { and v { (/ = 1,2) have finite energy, we obtain \hdμ 1 = \hdμ 2